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Existence and regularity for the FitzHugh-Nagumo equations with inhomogeneous boundary conditions. (English) Zbl 0712.35048

The author considers the FitzHugh-Nagumo equations which model the transmission of electrical impulses in a nerve axon. Existence and uniqueness of solutions to an initial-boundary value problem with irregular data is proved by the use of a Galerkin method. It is shown that this solution has a higher regularity if the data are smooth and satisfy compatibility conditions.
Reviewer: J.Sprekels

MSC:

35K57 Reaction-diffusion equations
92C05 Biophysics
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Hastings, S. P., Some mathematical problems from neurobiology, Am. math. Mon., 82, 881-895 (1975) · Zbl 0347.92001
[3] Jerome, J. W., Convergence of successive iterative semidiscretizations for FitzHugh-Nagumo reaction diffusion systems, S.I.A.M. J. numer. Analysis, 17, 192-206 (1980) · Zbl 0434.65095
[4] Lions, J. L., Quelques Methodes de Resolution des Problem aux Limits Non Lineaires (1969), Dunod Gauthier-Villars: Dunod Gauthier-Villars Paris · Zbl 0189.40603
[5] Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Vol. I (1972), Springer: Springer New York · Zbl 0223.35039
[6] Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Vol. II (1972), Springer: Springer New York · Zbl 0223.35039
[7] Peskin, Partial Differential Equations in Biology (1975), Courant Institute of Mathematical Sciences: Courant Institute of Mathematical Sciences New York · Zbl 0329.35001
[8] Rauch, J., Global existence of the FitzHugh-Nagumo equations, Communs partial diff. Eqns, 1, 609-621 (1976)
[9] Rauch, J.; Smoller, J., Qualitative theory of the FitzHugh-Nagumo equations, Adv. Math., 27, 12-44 (1978) · Zbl 0379.92002
[10] Schonbek, M. E., Boundary value problems for the Hugh-Nagumo equations, J. diff. Eqns, 30, 119-147 (1978) · Zbl 0407.35024
[11] Schwan, H. P., Biological Engineering (1969), McGraw-Hill: McGraw-Hill New York
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